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Solving systems of non‐linear equations with both free and positive variables
Author(s) -
Burns Scott A.,
Mueller Keith M.
Publication year - 1999
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19991230)46:12<1987::aid-nme750>3.0.co;2-l
Subject(s) - monomial , spurious relationship , sign (mathematics) , mathematics , nonlinear system , dimension (graph theory) , variables , mathematical optimization , mathematical analysis , pure mathematics , statistics , physics , quantum mechanics
A numerical method is presented for solving systems of non‐linear equations that contain some variables that are strictly positive and others that have no restriction on sign. Naturally positive variables arise frequently when modelling the behaviour of engineering systems, such as physical dimension, concentration of a chemical species, duration of an event, etc. When modelling systems of this type, it is also common to introduce additional variables that are not restricted in sign, such as stresses, displacements, velocities, accelerations, etc. Many numerical methods may experience performance difficulties due to the existence of spurious solutions which have negative components for one or more of the positive variables. Recently, the monomial method has been developed as an effective tool for systems with variables that are all strictly positive. This paper presents a hybrid method, combining the monomial method and Newton's method, for systems containing both types of variables. It is demonstrated that this hybrid method can be more effective in solving systems of equations with both positive and free variables than either method alone. Basins of attraction constructions are presented as a demonstration of the effectiveness of the hybrid method as applied to the design of a civil engineering frame structure. Copyright © 1999 John Wiley & Sons, Ltd.